Intermediate Algebra (11th edition)

(Marvins-Underground-K-12) #1

(d)


We could have used the method of substitution here.


(e)


Factor the difference of squares.
Factor the difference of squares again.
NOW TRY

= 1 x^2 + 921 x+ 321 x- 32


= 1 x^2 + 921 x^2 - 92


x^4 - 81


= 19 k+a+ 2219 k- a- 22


81 k^2 - 1 a+ 222 = 19 k 22 - 1 a+ 222 = 19 k+a+ 2219 k- 1 a+ 222


x^2 - y^2 = 1 x + y 2 1 x - y 2

334 CHAPTER 6 Factoring


⎩⎨⎧ ⎩⎨⎧


CAUTION Assuming that the greatest common factor is 1, it is not possible


to factor (with real numbers) a sum of squaressuch as in Example 1(e).In


particular, x^2 +y^2 Z 1 x+y 22 ,as shown next.


x^2 + 9


OBJECTIVE 2 Factor a perfect square trinomial. Two other special products


from Section 5.4lead to the following rules for factoring.


Perfect Square Trinomial

x^2  2 xyy^2  1 xy 22


x^2  2 xyy^2  1 xy 22


Because the trinomial is the square of it is called a perfect


square trinomial.In this pattern, both the first and the last terms of the trinomial


must be perfect squares. In the factored form twice the product of the first


and the last terms must give the middle term of the trinomial.


Perfect square trinomial; Not a perfect square trinomial;
middle term would have to be
and 212 m 2152 = 20 m. 16 por - 16 p.

4 m^2 = 12 m 22 , 25= 52 ,

4 m^2 + 20 m+ 25 p^2 - 8 p+ 64


1 x+y 22 ,


x^2 + 2 xy+y^2 x+ y,


NOW TRY
EXERCISE 1
Factor each polynomial.


(a)


(b)


(c)


(d)v^4 - 1


1 a+b 22 - 25

9 x^2 - 729

4 m^2 - 25 n^2

NOW TRY ANSWERS



  1. (a)
    (b)
    (c)
    (d) 1 v 2 + 121 v+ 121 v- 12


1 a+b+ 521 a+b- 52

91 x+ 921 x- 92

12 m+ 5 n 212 m- 5 n 2

Factoring Perfect Square Trinomials

Factor each polynomial.


(a)


Here, and The sign on the middle term is , so if


is a perfect square trinomial, the factored form will have to be


Determine twice the product of the two terms to see if this is correct.


This is the middle term of the given trinomial.


144 p^2 - 120 p+ 25 factors as 112 p- 522.


2112 p 21 - 52 = - 120 p


112 p- 522.


144 p^2 - 120 p+ 25


144 p^2 = 112 p 22 25 = 52. -


144 p^2 - 120 p+ 25


EXAMPLE 2

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